The Double Root

When the discriminant is zero, the equation has one solution. Take $x^{2}-4x+4 = 0$. The left side is a perfect square trinomial, so you can write it as $\left(x-2\right)^{2} = 0$. A square is zero only when the number being squared is zero, so $x-2 = 0$ and $x = 2$. There is one solution.

You can also factor the left side as $\left(x-2\right)\left(x-2\right) = 0$. Both factors give the same equation, $x-2 = 0$, so both give the same solution. A solution that comes from two equal factors is called a double root. The double root of this equation is 2.

The discriminant of $x^{2}-4x+4 = 0$ is also zero. Here $b^{2}$ is 16 and $4ac$ is 16, so the discriminant is $16-16 = 0$. Because it is zero, the plus and the minus in the quadratic formula give the same value, $x = 2$. The formula, the factoring and the squared form all agree that the equation has one solution.